Class 09

Late Policy

“This homework is due by 6:00pm on Friday, 9/20/24. No credit will be lost for assignments received by 7:00pm to account for issues with uploading. 10% of the points will be deducted from assignments received by 9:00am on Saturday, 9/21/24. Assignments turned in after this point are only eligible for 50% credit, so it benefits you to turn in whatever you have completed by the due date.”

Chapter 3 Objectives

• Define probability, random processes, and the law of large numbers

• Describe the sample space for disjoint and non-disjoint outcomes

• Calculate probabilities using the General Addition and Multiplication Rules

• Create a probability distribution for disjoint outcomes

Defining Probability

What does the word probability mean to you?

People interpret probability differently

Did your estimate fall within these ranges? Are these ranges reasonable?

So what is probability?

• A random process is one where you know which outcomes are possible (i.e. the sample space) but you don’t know which outcome comes next

• Examples of a random process: coin toss, die roll, stock market

How do you know a process is random?

Both Apple and Spotify took steps to make their “shuffle” features less random after complaints from users.

What went wrong?

• The human brain is good at finding patterns in noise, even when there are none

• If an artist is repeated “too soon”, the listener doesn’t feel the order is random

• We perceive a “random” distribution as also being “uniform” and “fair”

Example: Spotify Playlists

What if shuffle was truly random?

Each time the song changes, every song on the playlist is eligible to be played next

• Does not matter if the song was just played

• Does not matter who the artist is

We call this sampling with replacement.

• Like drawing a playing card, looking at it, then putting it back in the deck before the next draw

• Repetition of outcomes is possible

Sampling With Replacement

How often were artists repeated during Spotify’s original shuffle?

How often were artists repeated during Spotify’s original shuffle?

What is the probability of hearing a song by Chappell Roan?

• There is some “true” real-world probability that the next song is by Chappell Roan

  • Population proportion (\(p\))

• There is our “observed” probability that the next song is by Chappell Roan

  • Sample proportion (\(\hat{p}_n\))

Defining the sample space

The sample space \(s\) or \(S\) is the total collection of possible outcomes or events for a random process.

• Die rolls: 1, 2, 3, 4, 5, 6

• Coin flips: heads, tails

• Stock market: up, down, no change

For this example, the sample space could be all the songs on the playlist (n = 50) or all the artists who perform them (n = 26).

Example sample spaces

Another sample space

Representing the sample space - 1 event

Representing the sample space - complements

In the sample space \(S\), the complement of event A occurring is event A not occurring. This is written as A^C or A’.

2 Events - Disjoint Outcomes

Outcomes are disjoint or mutually exclusive if they cannot both happen at the same time

• Taylor Swift and Adele did not collaborate on any songs on this playlist

• The next song played can either be by Taylor Swift OR by Chappell Roan but not by Taylor Swift AND Chappell Roan

• The events “The next song is by Taylor Swift” and “The next song is by Chappell Roan” are disjoint/mutually exclusive

Defining 2 disjoint events in the sample space

2 Events - Non-Disjoint Outcomes

Non-disjoint outcomes can occur at the same time.

• Charli xcx and Billie Eilish collaborated on a song on this playlist
• The next song played could be by Charli xcx or by Billie Eilish OR by BOTH Charli xcx and Billie Eilish.
• The events “The next song is by Charli xcx” and “The next song is by Billie Eilish” are non-disjoint

Defining 2 non-disjoint events in the sample space

Example: Non-Disjoint Outcomes

The beaverduck from Tenso Graphics

Calculating probabilities

• Probabilities are proportions, or the number of observations with a particular value divided by…

  • the total number of observations in a sample (\(n\)) for the sample proportion (\(\hat{p}_n\))

  • the total number of outcomes in the sample space (\(s\)) for the population proportion (\(p\))

• Proportions range from 0 (no observations/outcomes) to 1 (all observations/outcomes)

• Also may be a percentage, ranging from 0% to 100% (multiply proportion by 100)

Probability notation and calculation

\[ \operatorname{Probability}(\operatorname{Event A})=\operatorname{P}(\operatorname{A}) \]

\[ \begin{aligned} \operatorname{Sample Probability}(\operatorname{A})&=\frac{\operatorname{count}(\operatorname{observation = A})}{\operatorname{count}(\operatorname{observations in sample})} \\ &=\hat{p}_n \end{aligned} \]

\[ \begin{aligned} \operatorname{Population P}(\operatorname{A})&=\frac{\operatorname{count}(\operatorname{event = A})}{\operatorname{count}(\operatorname{events in sample space})} \\ &=p \end{aligned} \]

Calculating Probabilities with Complements

\[ \begin{aligned} \operatorname{P}(\operatorname{S})&=1 \\ &= \operatorname{P}(\operatorname{A})+\operatorname{P}(\operatorname{A}^{\operatorname{C}}) \\ &= \operatorname{P}(\operatorname{A})+\operatorname{P}(\operatorname{A}^{\operatorname{'}}) \end{aligned} \]

Proportion of Songs by Artist (Population Probability)

Calculating the Population Probability (\(p\)) for a Single Event

\(p=\operatorname{Population Probability}(\operatorname{Next Song by Chappell Roan})\)

• The sample space for the population probability that the next song is by Chappell Roan when there is “true” shuffle or sampling with replacement is all songs on the playlist (\(n=50\)).

• Chappell Roan has 7 songs on the playlist, so the event “The next song is by Chappell Roan” occurs 7 times within the sample space.

Calculating the Population Probability (\(p\)) for a Single Event

The population probability \(p\) of the next song being by Chappell Roan is…

\[ \begin{aligned} p&=\operatorname{P}(\operatorname{Next Song by Chappell Roan}) \\ &=\frac{\operatorname{count}(\operatorname{Songs By Chappell Roan})}{\operatorname{count}(\operatorname{Total Possible Songs})} \\ &=\frac{7}{50} \\ &=0.14 \\ &= 14\% \end{aligned} \]

Proportion of Chappell Roan songs

Calculating the Sample Probability (\(\hat{p}_n\)) for a Single Event

• The sample space for the sample probability of the next song being by Chappell Roan when there is “true” shuffle or sampling with replacement is the number of songs listened to so far (\(n=1+\)).

• Each time a Chappell Roan song is played, an event is counted / recorded.

Calculating the Sample Probability (\(\hat{p}_n\)) for a Single Event

The population probability of the next song being by Chappell Roan is…

\[ \begin{aligned} \hat{p}_n&=\operatorname{P}(\operatorname{Next Song by Chappell Roan}) \\ &=\frac{\operatorname{count}(\operatorname{Songs Heard By Chappell Roan})}{\operatorname{count}(\operatorname{Total Songs Heard})} \\ &=\frac{x}{n} \end{aligned} \]

How well does a sample proportion represent the population proportion?

Should we listen to 1 song?

How well does the sample proportion represent the population proportion?

Should we listen to 5 songs?

How well does the sample proportion represent the population proportion?

Should we listen to 10 songs?

How well does the sample proportion represent the population proportion?

Should we listen to 100 songs?

How well does the sample proportion represent the population proportion?

Should we listen to 200 songs?

Law of Large Numbers

How well the sample proportion \(\hat{p}_n\) represents the population proportion \(p\) depends on the size of the denominator.

As more observations are collected, the sample proportion \(\hat{p}_n\) of a particular outcome approaches the population proportion \(p\) of that outcome.

Small Sample Size / Small Denominator = Unreliable

The sample proportion is an unreliable estimator of the population proportion when the sample size is small.

Large Sample Size / Small Denominator = Reliable Sample Proportion

The sample proportion is a reliable estimator of the population proportion when the sample size is large.